Math 140 Exam #1 by Professor Berg, Sep 30, 1992 - Prof. Kenneth R. Berg, Exams of Calculus

Download Math 140 Exam #1 by Professor Berg, Sep 30, 1992 - Prof. Kenneth R. Berg and more Exams Calculus in PDF only on Docsity! Math 140 Exam #1 Sep 30,1992 Professor: Berg Assistants: Al-Khal, Gurski, Li Your name Instructions: Circle the name of your assistant on the front sheet. Write your own name at the top of every sheet. Work each problem in the space provided (if more space is needed, work on the back of the same sheet, and clearly indicate that such work is to be considered as part of your solution). When calculation is needed, you must show that calculation as part of your solution. If such calculation is not shown, an answer may not be given credit even if it is correct. Present your work as clearly as you can. An incorrect answer may receive partial credit if the approach is correct and if we can figure out what you have done. Caution: You must neither give nor receive help of any kind from another student during this exam. If a question arises, summon a proctor. l.(30 points, 15 per part) Calculate the exact value of each of the following limits: r a. lim-+4 x — 4 b. limx->o 3z2 2. (10 points) Suppose that / is a function defined on an open interval, that a is a point within that interval, and that L is a number. Give the formal definition ( use e and 6) of the statement limx_a /(.r) = L. Your name 3. (36 points, 12 per part) a. Calculate lim [15 — 1x\ X —^O~*" ( [x] denotes the largest integer that does not exceed x.) Explain your answer with a sentence beginning: If x is near three but slightly larger than three then ... 1 — cosx 1 1 —cosSz b. It is a fact that hm ——~—— = -• Evaluate lim ———=—— z->o x2 2 i-»o 4x2 c. It is a fact that lim (cos x)~ l ' x exists. Use your calculator to estimate the value of this i—>o limit, correct to two decimal places.